Evaluation of Bootstrap Confidence Intervals Using a New Non-Normal Process Capability Index

PurposeIn order to develop a feasible and efficient method to acquire the long‐run availability of a parallel system with distribution‐free up and down times, the purpose of this paper is to perform the simulation comparisons on the interval estimations of system availability using four bootstrapping methods.Design/methodology/approachBy using four bootstrap methods; standard bootstrap (SB) confidence interval, percentile bootstrap (PB) confidence interval, bias‐corrected percentile bootstrap (BCPB) confidence interval, and bias‐corrected and accelerated (BCa) confidence interval. A numerical simulation study is carried out in order to demonstrate performance of these proposed bootstrap confidence intervals. Especially, we investigate the accuracy of the four bootstrap confidence intervals by calculating the coverage percentage, the average length, and the relative coverage of confidence intervals.FindingsAmong the four bootstrap confidence intervals, the PB method has the largest relative coverage in most situations. That is, the PB method is the best one made by practitioners who want to obtain an efficient interval estimation of availability.Originality/valueIt is the first time that the relative coverage is introduced to evaluate the performance of estimation method, which is more efficient than the existing measures.

In this work, we provide five different classical estimation methods for the gamma distribution and unknown parameters, which are used to estimate the newly constructed process capability index (PCI) C c C_{c} . Next, we use the aforementioned estimation techniques to consider five bootstrap confidence intervals: standard bootstrap (SB), percentile bootstrap (PB), student’s 𝑡 bootstrap (STB), bias-corrected percentile bootstrap (BCPB), and bias-corrected accelerated bootstrap (BCAB). These allow us to construct confidence intervals of the PCI C c C_{c} . Using the suggested techniques of estimate for small, moderate, and large sample sizes, Monte Carlo simulations are run to compare the performances of the bootstrap confidence intervals’ average widths and coverage probabilities. Simulation results indicate that MLE gives the least AW for all BCIs (SB, PB, BCPB, TB, and BCAB) when compared to all other estimation approaches evaluated in this paper. Further, simulation results demonstrate that, for almost all sample sizes and all of the examined estimation approaches, BCAB CI generates smaller AW and larger CP than their counterparts. Finally, for illustrative purposes, three real data sets are analysed.

The bias-corrected bootstrap confidence interval (BCBCI) was once the method of choice for conducting inference on the indirect effect in mediation analysis due to its high power in small samples, but now it is criticized by methodologists for its inflated type I error rates. In its place, the percentile bootstrap confidence interval (PBCI), which does not adjust for bias, is currently the recommended inferential method for indirect effects. This study proposes two alternative bias-corrected bootstrap methods for creating confidence intervals around the indirect effect: one originally used by Stine (1989) with the correlation coefficient, and a novel method that implements a reduced version of the BCBCI's bias correction. Using a Monte Carlo simulation, these methods were compared to the BCBCI, PBCI, and Chen and Fritz (2021)'s 30% Winsorized BCBCI. The results showed that the methods perform on a continuum, where the BCBCI has the best balance (i.e., having closest to an equal proportion of CIs falling above and below the true effect), highest power, and highest type I error rate; the PBCI has the worst balance, lowest power, and lowest type I error rate; and the alternative bias-corrected methods fall between these two methods on all three performance criteria. An extension of the original simulation that compared the bias-corrected methods to the PBCI after controlling for type I error rate inflation suggests that the increased power of these methods might only be due to their higher type I error rates. Thus, if control over the type I error rate is desired, the PBCI is still the recommended method for use with the indirect effect. Future research should examine the performance of these methods in the presence of missing data, confounding variables, and other real-world complications to enhance the generalizability of these results.

A confidence interval is an interval estimate of a parameter of a population calculated from a sample drawn from the population. Bootstrapping method, which involves producing several new data sets that are resampled from the original data in order to estimate parameter for each newly created data set, allowing an empirical distribution for the parameter to be estimated. Since certain statistics are harder to estimate, confidence intervals are rarely employed. Several statistics might necessitate multi-step formulas assuming that are impractical for calculating confidence intervals. This paper reviews research on the concept of bootstrapping and bootstrap confidence interval. The current narrative analysis was developed to answer the main research question: (1) What is the concept of the bootstrap method and bootstrap confidence interval? (2) What are the methods of bootstrapping to obtain confidence interval? This study has found general bootstrap method idea, various techniques of bootstrap methods, its advantages and disadvantages, and its limitations. There are normal interval method, percentile bootstrap method, basic method, first-order normal approximation method, bias-corrected bootstrap, accelerated bias-corrected bootstrap and bootstrap-t method. This study concludes that the advantages of using bootstrap CI is that it does not require any assumptions about the shape of distribution and universality of the approach. Bootstrapping is a computer-intensive statistical technique that relies significantly on modern high-speed digital computers to do massive computations.

This paper introduces a new capability index <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>C</m:mi> <m:mi>QT</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(C_{\rm QT})} which is based on an asymmetric loss function named Quadratic-Type Squared Error Loss Function (QTSELF). To estimate the proposed index <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>C</m:mi> <m:mi>QT</m:mi> </m:msub> </m:math> {C_{\rm QT}} , we have used five different classical methods of estimation (namely maximum likelihood, least squares and weighted least squares, Cramér–von-Mises, Anderson–Darling and maximum product of spacing) and their performances are compared. Additionally, three bootstrap confidence intervals, namely standard bootstrap ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒮</m:mi> </m:math> {\mathcal{S}} -boot), percentile bootstrap ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒫</m:mi> </m:math> {\mathcal{P}} -boot) and Student’s t bootstrap ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒯</m:mi> </m:math> {\mathcal{T}} -boot) are constructed. Their performances are compared in terms of average widths and coverage probabilities through a Monte-Carlo simulation study. Finally, two datasets are taken from electronic industries to demonstrate the practical utilities of the proposed estimation techniques and confidence intervals.

Confidence intervals of mean response time for an M/G/1 queueing system: Bootstrap simulation

The study focused on estimating the reliability index using confidence interval by comparing the Fishers Z and the bootstrap confidence interval methods. The rational for the study was to examine the bootstrap and the Fisher –Z methods and finding out the better of the two. The population of the study consists of the senior secondary school students in Egor local government area, Edo State. There are a total of 12 school with 8,207 students. A sample size of 410 representing 5% of the total population of students were randomly selected from the 12 schools. The instrument for data collection was the Open Hemisphere Brain Dominance Scale 1.0 (OHBDS) a personality scale designed by Eric Jorgenson (2015). It was adapted for the study. The instrument was validated. The reliability was part of the issues raised in the study. The data were analyzed using the Pearson Product-Moment Correlation Coefficient to determine the reliability. The Fishers Z 95% and the Bootstrap (percentile and bias corrected and accelerated) confidence interval were also used. The findings revealed that as the sample size became large, the length of the interval became narrower; the three methods utilized in this study yielded the same length of the interval (width) when the same size was 100 and 150; and as the sample size increases, the bias corrected and accelerated bootstrap gave a shorter interval length, thereby becoming the best of the three method considered in the study. It was therefore recommended that reporting reliability should be based on interval estimation as against the point estimate, the sample size should be at least 100 and the bootstrap confidence interval should be adopted as it is not liable to the normality condition associated with the classical statistics.

On the bootstrap confidence intervals of the process incapability index Cpp

The generalized Pareto distribution (GPD) is a two-parameter family of distributions which can be used to model exceedances over a threshold. We compare the empirical coverage of some standard bootstrap and likelihood-based confidence intervals for the parameters and upper p-quantiles of the GPD. Simulation results indicate that none of the bootstrap methods give satisfactory intervals for small sample sizes. By applying a general method of D. N. Lawley, correction factors for likelihood ratio statistics of parameters and quantiles of the GPD have been calculated. Simulations show that for small sample sizes accuracy of confidence intervals can be improved by incorporating the computed correction factors to the likelihood-based confidence intervals. While the modified likelihood method has better empirical coverage probability, the mean length of produced intervals are not longer than corresponding bootstrap confidence intervals. This article also investigates the performance of some bootstrap methods for estimation of accuracy measures of maximum likelihood estimators of parameters and quantiles of the GPD.

In this article, we consider six process capability indices (PCIs) whose quality characteristics have a normal distribution, out of which first PCI 𝒞 p {\mathcal{C}_{p}} was developed in [J. M. Juran, Juran’s Quality Control Handbook, 3rd ed., McGraw-Hill, New York, 1974], next PCI 𝒞 p ⁢ m {\mathcal{C}_{pm}} was developed in [T. C. Hsiang and G. Taguchi, A tutorial on quality control and assurance, Annual Meeting on the American Statistical Association, Las Vegas 1985], third PCI 𝒞 p ⁢ m ⁢ c {\mathcal{C}_{pmc}} was developed in [M. Saha, S. Dey and L. Wang, Parametric inference of the loss based index C p ⁢ m C_{pm} for normal distribution, Qual. Reliab. Eng. Int. 38 2022, 10.1002/qre.2987], fourth PCI 𝒞 p ⁢ m ′ {\mathcal{C}^{\prime}_{pm}} was developed in [S. Dey, W. Wang and M. Saha, Modified estimation and confidence intervals of an asymmetric loss based process capability index 𝒞 p ⁢ m ′ \mathcal{C}^{\prime}_{pm} , Qual. Reliab. Eng. Int. 38 2022, 8, 4033–4048], fifth PCI 𝒞 p ⁢ c {\mathcal{C}_{pc}} and sixth one 𝒞 p ⁢ m ⁢ c ′ {\mathcal{C}^{\prime}_{pmc}} are newly proposed in the paper. Next, we estimate the cited process capability indices using the method of moment estimation (MOM) approach when the underlying process follows normal distribution. A simulation study is conducted to evaluate the performance of these indices with respect to their mean squared errors. Additionally, confidence intervals are constructed using the asymptotic confidence interval (ACI), the generalized confidence interval (GCI) and parametric bootstrap confidence interval (BCI). Using Monte Carlo simulation, the performance of the GCI and BCI is compared in terms of average width and associated coverage probabilities. Finally, four data sets, two related to electronic industries and two related to food industries are re-analyzed to show the applicability’s of the suggested indices, MOM estimation, GCI and BCI.

The Poisson distribution is commonly applied in statistical modeling to represent the count of events, assuming that the events are independent and occur at a constant rate. This assumption, however, may not always hold true in real-life situations. The Poisson distribution may be inadequate if the underlying rate of occurrence is not constant. The mixed Poisson distribution is proposed to solve this limitation because it permits the rate parameter of the Poisson distribution to be random rather than fixed. The Poisson-Pranav distribution, which is classified as a type of mixed Poisson distribution, has been commonly utilized to analyze count data that exhibits over-dispersion over many domains. However, no research has been done into building confidence intervals for the parameter of the Poisson-Pranav distribution using the bootstrap method. Using Monte Carlo simulation, the coverage probabilities and average lengths of the percentile bootstrap (PB), basic bootstrap (BB), and biased-corrected and accelerated (BCa) bootstrap’s interval-estimation performances were compared. The bootstrap method was not appropriate for achieving the desired nominal confidence level with small sample sizes. In addition, the performance of the bootstrap confidence intervals did not differ significantly when the sample size was increased considerably. In each case studied, the BCa bootstrap confidence intervals performed better than the others. The effectiveness of bootstrap confidence intervals was demonstrated by applying them to the number of COVID-19 deaths in Belgium. The computations substantially supported the proposed bootstrap confidence intervals.

Process capability indices (PCIs) are most effective devices/techniques used in industries for determining the quality of products and performance of manufacturing processes. In this article, we consider the process capability index which is based on an asymmetric loss function (linear exponential) and is applicable to normally as well as non-normally distributed processes. In order to estimate the PCI when the process follows exponentiated exponential distribution, we have used ten classical methods of estimation and the performances of these classical estimates for the index are compared in terms of mean squared errors (MSEs) through simulation study. Also, the confidence intervals for the index are constructed based on four bootstrap confidence interval (BCIs) methods. A simulation study is performed in order to compare the performance of these four BCIs in terms of average width and coverage probabilities. We use two published data sets related to electronic and food industries to illustrate the performance of the proposed methods of estimation and BCIs. All the data sets show that width of bias-corrected accelerated bootstrap interval is the lowest among all other considered BCIs.

The data used for the analysis were collected from multiple regions or years. Evaluating each region or year separately may be insufficient for drawing comprehensive inferences and may fail to reveal statistically significant differences. To ensure the reliability of the analysis and to enable overall conclusions, it is necessary to apply a statistical method known as simultaneous confidence intervals. This technique enables the simultaneous construction of confidence intervals for multiple parameters. Therefore, we proposed and evaluated methods for constructing simultaneous confidence intervals for all pairwise differences between the coefficients of variation in zero-inflated Birnbaum-Saunders distributions. The methods utilized for constructing simultaneous confidence intervals comprise the generalized confidence interval (GCI), the bootstrap confidence interval (BCI), the method of variance estimates recovery (MOVER), the MOVER based on GCI, the MOVER based on BCI, the Bayesian credible interval, and the highest posterior density interval (HPD). Monte Carlo simulations were employed to evaluate the performance of each method, which involved the assessment of coverage probabilities and average widths under a set of parameter configurations and sample sizes. The generalized confidence interval method was the most efficient overall, as indicated by the simulation results. Finally, all proposed methods were applied to real-world wind speed data to examine their practical applicability and to demonstrate the consistency of the results between the simulation study and real-world applications.

Theory: Bootstrapping is a nonparametric approach to statistical inference that relies on large amounts of computation rather than mathematical analysis and distributional assumptions of traditional parametric inference. It has been shown to provide asymptotically accurate inferences for a wide variety of statistics. Hypothesis: Bootstrapping may make more accurate inferences than the parametric approach under two general circumstances: 1) when the assumptions of parametric inference are not tenable, and 2) when no parametric alternative exists for a problem. Methods: Monte Carlo simulation is used to test the performance of bootstrap and parametric confidence intervals for both types of situations in which bootstrapping is hypothesized to be superior to parametric inference. A single data example is used to illustrate the use of the bootstrap: a seats/votes model of U.S. House elections from 1932 to 1988. Results: My central conclusions are that in the cases examined: 1) bootstrap confidence intervals are at least as good as the parametric confidence interval and sometimes better, 2) OLS parametric confidence intervals do not perform too badly when the model error is non-normal, especially as sample size increases, and 3) when no parametric alternative exists, the bootstrap provides a reasonable method of making statistical inferences.

Since sampling variation would lead to the inaccurate assessment of process capability indices (PCIs), the interval estimation of PCIs has received considerable attention recently. The coverage probabilities (CPs) of the widely used bootstrap confidence intervals (BCIs) of PCIs are not sufficiently close to their nominal confidence level. Moreover, the bootstrap method is time‐consuming. This paper develops a procedure for constructing generalized confidence intervals (GCIs) of two widely used percentile‐based PCIs for the Birnbaum–Saunders (BS) distribution. A simulation study is conducted and the results indicate that the proposed GCI outperforms its bootstrap counterparts in terms of the CPs, the average widths (AWs) of the confidence intervals, and the variability of the interval widths. Finally, two real examples are used to illustrate the implementation of the proposed procedure.